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Star-multiplicative graphs in pointed protomodular categories
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| Protomodularity, in the pointed case, is equivalent to the Split Short Five Lemma. It is also well known that this condition implies that every internal category is in fact an internal groupoid. In this work, this is condition (II) and we introduce two other conditions denoted (I) and (III). Under condition (I), every multiplicative graph is an internal category. Under condition (III), every star-multiplicative graph can be extended (uniquely) to a multiplicative graph, a problem raised by G. Janelidze in [10] in the semiabelian context. When the three conditions hold, internal groupoids have a simple description, that, in the semiabelian context, correspond to the notion of internal crossed module, in the sense of [10]. | 281.95 KB | Adobe PDF |
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Orientador(es)
Resumo(s)
Protomodularity, in the pointed case, is equivalent to the Split Short Five Lemma. It is also well known that this condition implies that every internal category is in fact an internal groupoid. In this work, this is condition (II) and we introduce two other conditions denoted (I) and (III). Under condition (I), every multiplicative graph is an internal category. Under condition (III), every star-multiplicative graph can be extended (uniquely) to a multiplicative graph, a problem raised by G. Janelidze in [10] in the semiabelian context. When the three conditions hold, internal groupoids have a simple description, that, in the semiabelian context, correspond to the notion of internal crossed module, in the sense of [10].
Descrição
Fonte: http://www.tac.mta.ca/tac/volumes/23/9/23-09.pdf
Palavras-chave
Internal category internal groupoid re°exive graph multiplicative graph star-multiplicative graph jointly epic pair admissible pair jointly epic split extension split short ¯ve lemma pointed protomodular
Contexto Educativo
Citação
Martins-Ferreira, N. (2010). Star-multiplicative graphs in pointed protomodular categories. Theory and Applications of Categories, 23(9), 170-198. DOI: https://doi.org/10.70930/tac/iodb71v5.
Editora
Mount Allison University
Licença CC
Sem licença CC